Scattering of a plane gravitational wave by a magnetic dipole field in the Schwarzschild metric

(1)

A NNALES DE L ’I. H. P., SECTION A

T HOMAS E LSTER

Scattering of a plane gravitational wave by a magnetic dipole field in the Schwarzschild metric

Annales de l’I. H. P., section A, tome 34, n^o2 (1981), p. 163-172

<http://www.numdam.org/item?id=AIHPA_1981__34_2_163_0>

L’accès aux archives de la revue « Annales de l’I. H. P., section A » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.

org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

(2)

163

Scattering ^of

^a

plane gravitational

^wave

by

^a

magnetic dipole ^field

in

the Schwarzschild metric

Thomas ELSTER

Wissenschaftsbereich Relativistische Physik, ^SektionPhysik, Friedrich-Schiller-Universitat, ^DDR-69Jena, Max-Wien-Platz 1 Ann. Inst. Poincaré,

Vol. XXXIV, ^nO2, 1981,

Section A : Physique ^’théorique. ^’

ABSTRACT. The equations governing ^thepropagation ^ofhigh-frequency coupled gravitational ^andelectromagnetic ^{waves are}integrated ^for^the^case

of a plane gravitational ^waveimpinging ôntoâSchwarzschild space-time equipped ^withâmagnetic dipole field. The generated electromagnetic ^wave

is focused due to the gravitational ^lensêffect.Ânexpression îsderived for the intensity ôf^theelectromagnetic ^waveⁱⁿ^{the focal}region. În^the^caseôf alignment of incidence direction and magnetic âxis^no^rotation^{of the}pola-

rization plane ^occurs.The relevance of the effect for the exterior region ^of

neutron stars is discussed.

1. INTRODUCTION

As is well known, electromagnetic ^andgravitational perturbations ^cannot

be treated separately ⁱⁿregions ôfspace-time occupied by âstrong êlectro- magnetic ^field.Rather, they ^lose^theirindividuality and appear only ⁱⁿâ coupled ^manner.^Thiscoupling ^leads^to^theeffect of conversion of gravita-

tional into electromagnetic ^waves(and vice versa). In the simplest ^case

Minkowski space-time ^{is taken}âsbackground ândthe conversion effect is studied by using external static ôrstationary electromagnetic ^fields[7-7C],

fields of moving charges [77, 12] ândrotating magnetic dipoles [l3] ôrby investigating bremsstrahlung processes (14]. Similarly, the effect of photo- production ôfgravitons in static electromagnetic fields is predicted by â quantized version of linearized general relativity [15, 16].

Annales de l’Institut Henri Poincaré - Section A - Vol. XXXIV, 0020-2339/1981/163/$ 5.00/

(Ç) Gauthier-Villars

(3)

If the background space-time ^iscurved, ^thelinearized Einstein-Maxwell

equations êxhibitâ^rathercomplicated structure. Therefore, ôneêither

chooses a special background ^metricor assumes the waves to be of small

wavelength. Âs^farâsthe former possibility îsconcerned, ^severaltechniques

were developed ^{to treat}perturbations of the Reissner-Nordstrom solution,

among them Zerilli’s approach [17], Moncrief’s method [18] ^based^{on a}

Hamiltonian formalism as well as the approach employing ^the^Newman-

Penrose [19] ^formalism(see ^the^recentreview article by ^Bicak[20] containing

a description ^andcomparison ^{of all}^thesetechniques). Using ^thesemethods,

a number of authors [21-28] calculated the efficiency ^of^theconversion effect in the Reissner-Nordstrom background. ^Inparticular, ^theconversion cross

section of charged ^black^holes^canbe derived [2~-2~]. Ôn^theôtherhand, high-frequency ^{waves can}^be^treatedâsperturbations away from arbitrary

solutions of the Einstein-Maxwell equations by deriving propagation ^equa-

tions governing ^thechange ^of^the^waveamplitudes along the rays [29-33].

The new effect which emerges is a Faraday-like ^rotation^{of the}polarization plane ^oflinearly polarized ^radiation^withrespect ^to^aparallely propagated

tetrad along ^therays.

The present paper differs from previous research in two ways. First, ^we

assume a somewhat more realistic situation by using ^aSchwarzschild solution

equipped ^with^amagnetic dipole field. In contrast to the commonly ^used

electric monopole fields, this situation _{may be}a good ^modelfor, ê.g., the exterior region ôfâneutron star. Second, ^we^takeîntoâccount^thefocusing

of the generated electromagnetic radiation due to the gravitational ^lens^effect.

In particular, ^theintensity ^{in the}^focalregion, ^which^can^be^obtainedby

means of wave optical ^methods[34-38], îsôf^someînterest.^The^mainquestion

which arises is whether the rather pessimistic predictions ^ofprevious papers

concerning ^theastrophysical significance ^ofthe conversion effect (see,

e. g., [IO]} ^{may be}changed ⁱⁿ^this^case.

The following ^section^outlines^{the basic}equations governing the propa-

gation ôfcoupled gravitational ândelectromagnetic ^wavesⁱⁿ^thehigh- frequency ^limitâs^derivedⁱⁿ(33]. This formalism is then used in sec. 3 and

sec. 4 to investigate ^thescattering process described above. We use units with c ⁼1 and the signature (+,+,+,2014). ~ denotes Einstein’s gravita-

tional constant.

2. PROPAGATION OF INTERACTING ELECTROMAGNETIC AND GRAVITATIONAL PERTURBATIONS

IN THE LIMIT OF SMALL WAVELENGTHS

In the approximation ^ofgeometrical optics, electromagnetic ^andgravi-

tational waves are conveniently ^studiedby ^meansôf^the^nulltetrad formalism introduced by ^Newmanand Penrose [19]. ^Considerâ^solution^{of the}Êinstein-

Annales de l’lnstitut Henri Poineare - Section A

(4)

165 SCATTERING OF A PLANE GRAVITATIONAL WAVE BY A MAGNETIC DIPOLE FIELD

Maxwell equations characterized by îtsWeyl ^tensor ⁺ ândîts

Maxwell tensor Fa~ ⁺ ^The^parameter^G^is^supposed^tobe small in order to ensure that the field equations ^canbe linearized. Let ø A’ ^and

the tetrad components ôf Fa~ ând ^with^respect^toânull tetrad defined in the background space-time. If the scale on which the perturbations

vary is much smaller than the typical ^curvatureradius of the background geometry, ^wavesmainly propagate along ^nullgeodesics ^of^thebackground metric, ^which^allowscalculations to be simplified by identifying ^{the tetrad}

vector la ^with^the^ray^vector^andassuming the remainder of the tetrad to

undergo parallel transport along ^therays. Under these assumptions ^the perturbations ^arecompletely characterized by ^the^twocomplex ^functionsq 4

and ~2, which take for nearly monochromatic waves the form

with l03B1 ⁼S ,a.. ^úJ^is^alarge parameter ensuring ^that^the^waveoscillates on a

small enough scale. Its upper limit is determined by ^theassumption ^{that the} leading contribution to the curvature tensor of the perturbed space-time originates ^{in the}background metric. The amplitudes describing right-

(0) (0) (0)_ (0)_

handed

(~4 , ~ 2 )

and left-handed (~4 , ~ 2 ) polarized ^{waves are}^invariant

with respect ^tocoordinate _gaugetransformations owing ^to^thehigh-fre-

quency assumption (1). ^Thepropagation equations

which determine the rate of change ^of^theamplitudes along ^therays, ^are

coupled by ^the^tetradcomponent ~o ^of^thebackground Maxwell field. In the following ^we^confine^ourselves^toright-handed polarized waves, since the corresponding equations describing left-handed polarized ^waves^do^not

differ essentially ^from^thosereferring ^to^thehelicity ^labelled^{+ .}The system (2 . 2) ^can^bedecoupled by introducing ^normal

modes(0)~ 1/2:

(1) Rather than project ^bothbackground parts and perturbations of the tensor fields onto a background ^nulltetrad, ^{we can}define the variables (2.1) equally ^wellby splitting the tetrad components (formed ^withâperturbed null tetrad) întounperturbed parts ând perturbations. În^this^case^theamplitudes ârealso invariant with respect ^totetrad _gauge transformations and, in particular, identical with the above-constructed amplitudes.

Vol.XXXIV.n°2- ^1981.

(5)

Inserting (2. 3) ^into(2.2) ^we^obtain^thesystem

which can immediately ^besolved, ^and^asystem ^for^theelements of the

(unitary) ^matrix^T+.Introducing ^the^new^variables’1 and y by

(y ^affineparameter), ^we^can^write^thelatter in the simple ^form

Although (2. 6) ^{is in}general ^acomplicated system, ^it^hassimple ^solutionsⁱⁿ important special ^cases[29-33].

3. A PLANE GRAVITATIONAL WAVE IMPINGING ONTO A SCHWARZSCHILD SPACE-TIME

In the remainder of this _{paper the}underlying space-time is assumed to be described by the Schwarzschild metric

Since (3.1) represents ^{a vacuum}^solution^of^the^Einsteinequations, ^we neglect ^{the back}^reaction^{of both}background Maxwell field and electro-

magnetic ^{wave on}the metric. As already pointed ^outby ^Zeldovich[10],

this leads to no inconsistencies if the change ^{of the}beating phase (defined below) along ^a^raytraversing ^thespace-time ^domainconsidered is small

compared ^withunity. ^Otherwise^oneencounters instabilities indicating ^that

the back reaction of F~ ^on~ ^cannot^beneglected. ^Moreover,^{in order}

to be compatible with the linearization of the field equations ^as^described above, ^thetypical ^{values of}^thebackground Maxwell field Fa~ ^are,ⁱⁿ^turn, supposed ^to^belarge compared ^with^thetypical ^{values of}^the^wave

Consider ^abundle of null geodesics each of which lies in ^ahypersurface

~p ⁼const of the metric (3.1). ^Anappropriate ^null^tetrad^whichundergoes

Annales de t’Institut Henri Poincaré - Section A

(6)

SCATTERING OF A PLANE GRAVITATIONAL WAVE BY A MAGNETIC DIPOLE FIELD 167

parallel propagation along ^§^these^"rays is given by (oc ⁼1, 2, 3, ⁴corresponds

with the abbreviations

The congruence is completely characterized if the impact parameter a ^is known as a function of rand 8. Although ^thespin coefficients associated with (3.2 a, b~ ^are^ratherlenthy expressions, ^weeasily succeed in solving ^the propagation equations ^sinceonly p is needed :

Using (3.3) ^we^obtain

where _yis an affine parameter ^definedalong ^therays. As mentioned above,

we are interested in an incident plane gravitational ^wavecoming ^{from the}

direction 8 = 7r. Hence the yet unspecified function f(a) ^has^tobe appro-

priately ^{fixed :}f (a) _ --~ 1/2’In â.Taking întoâccount_only^waves^with positive helicity ^weobtain with the help ôf(2.1), (2. 3), (2.4) ând(3.4) ^the perturbations

The coupling ^{of the}^two^kinds^of^wavesis determined by the appearance of

Vol. XXXIV, ^no^{2 - 1981.}

(7)

the beating phase Sb ⁼ ^The^constants^ofintegration

C12

should not depend ^on^theimpact parameter a, ^whichcorresponds ^to^the boundary condition of ^anincident plane ^wave.Moreover, ^{once a}^solution

of (2.6) ^{has been}fixed, ^the^constants ^mustbe chosen such that the

incoming ^wave^contains^noelectromagnetic part.

Consider a space-time domain where the ray ^vectorla ^exhibits^apositive

radial component (E1 - ⁺1), ^i. e., points ^off^thegravitating ^mass(this

appears ^tobe the case in the region ^{0 S :)} ^~cbehind the mass). ^In^this

domain the phase ^{of the}^wave^isgiven by

where YT ^denotes^the^classicalturning point ^{in the}^outerpart ^{of the}^effective

potential ^of^azero-rest-mass particle. Ignoring secondary rays, which revolve the deflecting ^mass^{in the}vicinity of the circle ~ ⁼3M before they

return to infinity, ^we^have^toconsider the two main _rayspassing ^the^massât opposite ^sides.^Theyet unspecified sign in front of the last term in (3.6) depends ôn^whichôf^these^two^raysîsconsidered. Therefore, ^theperturbations (3 . 5) âreⁱⁿgeneral ^sumsôf^twocontributions associated with these rays.

As mentioned in the Introduction, ^{we are}interested in the amplification

of the generated electromagnetic ^radiation^due^to^thegravitational ^lens

effect. Treatments of the gravitational focusing ^ofhigh-frequency ^radiation by the Schwarzschild field [34-38] suggest ^that^the^maximumintensity ^is^to

be expected ⁱⁿ^{a narrow}^focalregion around the axis of symmetry ~ ⁼^0.

In particular, ^{the focal}region ^ischaracterized by the condition r~2 ^«^2Min the case of an incident plane ^wave.Unfortunately, ^this^conditionjust

describes the range in which the expressions (3 . 5) ^fail^to^beapplicable. ^This

is not surprising since all rays which lie in various _~p⁼const surfaces and which have practically ^the^sameimpact parameter interfere in the vicinity

of the axis 8 ⁼0. In order to circumvent this difficulty, ^{we now}^remember

the exact waveoptical ^treatmentof the field in the focal region involving ^an expansion ⁱⁿ^terms^ofspherical ^harmonics[36-38]. For very small 8, ^the

contributions (3.5), (3.6) associated with the ^tworays differ only ⁱⁿâ 8-dependent factor. The ^sumof these factors yields â^cosfunction. As shown in previous papers, the actual perturbations âreobtained in the focal

region by making ^thesimple substitution

where Jo ^denotes^aBessel function of the first kind.

Annales de l’Institut Henri Poincare - Section A

(8)

169

SCATTERING OF A PLANE GRAVITATIONAL WAVE BY A MAGNETIC DIPOLE FIELD

In order to calculate the energy flux associated with both electromagnetic

and gravitational perturbations, ^the corresponding energy-momentum

tensors are needed. Adopting Isaacson’s j39] reasonable definition of effective energy and ^momentumcarried by ^a^shortgravitational ^{wave we}get the expressions

which are invariant with respect ^tocoordinate _gaugetransformations in this

approximation. Using (3.5)-(3.8) ônefinally ârrivesâtânexpression ^for

the intensity (magnitude ^{of the}time-averaged, three-dimensional Poynting vector) ^{of the}electromagnetic ^waveⁱⁿ^{the focal}region normalized to the

intensity of the incident gravitational ^{wave :}

indicates that the values of the matrix T+ have to be taken for the incident plane ^wave.

4. SCATTERING BY A MAGNETIC DIPOLE FIELD So far, nothing has been assumed about the electromagnetic background

field which enters into the beating phase Sb. ^For^the^{sake of}simplicity,

we restrict ourselves to the simplest non-spherically symmetric multipole,

i. _e.,a magnetic dipole. ^In^this^case^the^exactsolution of the Maxwell

equations ⁱⁿthe metric (3 .1 ) ^isgiven by

m is the magnitude ^{of the}(time-independent) dipole moment, and x ^denotes

the angle ^between^thedirection from which the incident gravitational ^wave

comes in (8 = 7r) ^{and the}magnetic axis. F is a hypergeometric ^{function :}

Vol. XXXIV, ^nO^{2 -}^1981.

(9)

Projection ^of(4.1) ^ontothe null tetrad {3 . 2 a, b) yields the tetrad component

which couples ^theperturbations ^{in the}^shortwaveapproximation. ^Theargu- ment ~ ôf4>0 ^{is in}general â^rathercomplicated ^functionôf^theaffine parameter y along âray. Consequently, ^thepolarization plane ôflinearly pola-

rized radiation does not undergo parallel transport, ^but^rotatesalong ^aray while _energyis partly transferred from gravitational ^toelectromagnetic

waves. Unfortunately, ^thesystem (2.6) ^does^not^seem^to^be^solvable^ana- lytically.

In order to get ^anestimate of the conversion effect despite ^thisdifficulty,

we now introduce two further simplifications. First, ^{we assume}alignment ^of

incidence direction and magnetic âxis(x ⁼0). În^thisspecial ^case^theargu- ment ~ ôf4>0 ^takes^the^constant^{value of}7c/2, ând(2 . 6) îs^solvedby â^constant

matrix :

Since the incoming plane ^wave^was^assumed^to^contain^noelectromagnetic part, ^we^have^toput

C1 -

Consequently, ^thepolarization plane ^of linearly polarized ^waves^does^not^rotatealong the rays. Second, ^we^assume

r » 2M in (3.9), ^whichimplies ^{that the}impact parameters ^{of the}^tworays

are large compared ^with^2M^as^{well. In}^thisapproximation ^{the null}geodesic equation ^is^solvedby

Using ^theexpansion

of the hypergeometric ^function^one^obtains^thebeating phase

on the axis of symmetry ~ ⁼^0.Assuming ^the^rateof conversion of gravita-

tional wave energy into electromagnetic ^waveenergy ^tobe small (i. e.,

Sb ^«1 ) ^weget ^from(3 . 9)

for ~ ^’ ⁼0. Thus the intensity of the induced electromagnetic ^wave^decreases

on 1 the axis of symmetry ⁱⁿradial direction as r - 2, ^since^thestrength ^of^the

Annales de l’lnstitut Henri Poincare - Section A

(10)

SCATTERING ^{OF A}PLANE GRAVITATIONAL WAVE BY A MAGNETIC DIPOLE FIELD 171

magnetic dipole field falls off with increasing ^distance^from^the^massacting

both ^asgravitational ^lens^andmagnetic dipole.

Unfortunately, ^even^underfavourable astrophysical conditions the

expected effect is rather small. Consider âneutron star of one solar mass, with a radius of 2’10~ ^mand âdipole-like magnetic ^fieldôf^1012Gât^the poles ôn^thesurface of the star. Due to the shadow effect of the deflector [36],

the focal region ^starts^at^a^distance^{of 6.9’}104m from the centre of the star.

If the intensity ôf^thegenerated electromagnetic ^waveâmounts^there^toône-

tenth of the intensity ^{of the}încidentgravitational ^wave,âgravitational wavelength ^{of about}2’ 10’~ ^mis required, ^whichîsextremely ^smallûnlike

the electromagnetic ^case.Though situations with much smaller rates of conversion are also important owing ^to^the^strongercoupling of electro-

magnetic ^radiation^tomatter, other phenomena ^influenceboth the conversion effect [10] ^{and the}focusing ^ofthe radiation [40, 41 ] .

Nevertheless, ît^would^{be of}^someînterest^to^treat^the^caseôfâmagnetic

axis which is inclined relative to the incidence direction and the case of a

rotating magnetic dipole ^field[42, 43] generated by ^apulsar. ^In^these^cases

the system (2.6) ^has^to^be^solvednumerically.

ACKNOWLEDGMENT

I am grateful ^to^Dr.^R.^A.^Breuer^forsending ^me^acopy of his paper [28]

prior ^to^itspublication.

[1] ^D.BOCCALETTI, ^V.^DESABBATA, ^P.FORTINI, ^C.GUALDI, ^NuovoCim., ^{t. B}70, 1970, p. 129.

[2] ^D.BOCCALETTI, ^F.OCCHIONERO, Lett. Nuovo Cim., ^t.2, 1971, p. 549.

[3] V. ^DESABBATA, ^P.FORTINI, ^C.GUALDI, L. FORTINI BARONI, Acta Phys. Pol., ^{t. B}5, 1974, p. 741.

[4] ^J.VINET, C. R. Acad. Sc. Paris, ^t.^A283, 1976, p. 1061.

[5] ^J.^VINET,^{C. R.}^Acad.^Sc.^Paris,^t.A 284, 1977, p. 827.

[6] ^{V. I.}DENISOV, ^Izv.Vyssh. ^Ucheb.^Zav.Fiz., No. 1, 1978, p. 137.

[7] ^{V. I.}DENISOV, ^Izv.Vyssh. ^Ucheb.^Zav.Fiz., No. 3, 1978, p. 152.

[8] ^S.BOUGHN, Phys. Rev., ^{t. D}11, 1975, p. 248.

[9] ^{W. K.}^DELOGI, ^A.^R.MICKELSON, Phys. Rev., ^t.^D16, 1977, p. 2915.

[10] ^{J. B.}ZELDOVICH, Zh. Eksp. ^Teor.Fiz., ^t.65, 1973, p. 1311; ^Sov.Phys.-JETP, ^t.38, 1974, p. ^652.

[11] ^M.SASAKI, ^H.SATO, Progr. ^Theor.Phys., ^t.60, 1978, p. 148.

[12] ^{W. K.}^DE^LOGI,^A.^R.MICKELSON, ^NuovoCim., ^t.^B47, 1978, p. 192.

[13] ^{V. I.}DENISOV, ^Zh.Eksp. ^Teor.Fiz., ^t.74, 1978, p. 401.

[14] ^R.A. MATZNER, ^{V. D.}SANDBERG, Phys. Rev., ^t.^D16, 1977, p. 2939.

[15] ^G.PAPINI, ^{S. R.}VALLURI, ^{Can. J.}Phys., ^t.53, 1975, p. 2306; ^t.53, 1975, p. 2312;

t. 53, 1975, p. 2315; t. 54, 1976, p. 76; t. 56, 1978, p. 801.

[16] ^G.PAPINI, ^{S. R.}VALLURI, Phys. Rep., ^t.^C33, 1977, p. 51.

Vol. XXXIV, ^nO^{2 - 1981.}

(11)

[17] ^{F. J.}ZERILLI, Phys. Rev., t. D 9, 1974, p. 860.

[18] ^V.MONCRIEF, Phys. Rev., ^t.D 12, 1975, p. 1526.

[19] ^{E. T.}^NEWMAN,^R.^PENROSE,^J.^Math.Phys., ^t.3, 1962, p. 566.

[20] J. BICÁK, ^Czech.^J.Phys., ^t.^B29, 1979, p. 945.

]21] ^{D. M.}CHITRE, ^{R. H.}PRICE, V. D. SANDBERG, Phys. ^Rev.Lett., ^t.31, 1973, p. 1018;

Phys. Rev., t. D 11, 1975, p. 747.

[22] ^M.JOHNSTON, ^R.RUFFINI, ^F.ZERILLI, Phys. ^Rev.Lett., ^t.31, 1973, p. 1317; Phys.

Lett., t. B 49, 1974, p. 185.

[23] ^{D. W.}OLSON, W. G. UNRUH, Phys. ^Rev.Lett., ^t.33, 1974, p. 1116.

24] G. A. ALEKSEEV, Dokl. Akad. Nauk SSSR, ^t.222, 1975, p. 312; Sov. Phys.-Dokl., t. 20, 1975, p. 336.

[25] ^{D. L.}^GUNTER,^Philos.^Trans.Roy. Soc., ^t.296, 1980, p. 497.

[26] ^R.^A.MATZNER, Phys. Rev., ^t.^D14, 1976, p. 3274.

[27] ^R.^FABBRI,^Nuovo^Cim.,^t.^B^{40, 1977,}^{p. 311.}

[28] ^R.^A.^BREUER,^M.ROSENBAUM, ^{M. P.}RYAN, R. A. MATZNER, Phys. Rev., ^D(sub- mitted).

[29] ^U.^H.^GERLACH,Phys. ^Rev.Lett., ^t.32, 1974, p. 1023.

[30] ^{U. H.}GERLACH, Phys. Rev., t. D 11, 1975, p. 2762.

[31] N. ^R.SIBGATULLIN, ^Zh.Eksp. ^Teor.^Fiz.,^t.66, 1974, p. 1187; Sov. Phys.-JETP,

t. 39, 1974, p. 579.

[32] ^T.TOKUOKA, Progr. Theor. Phys., ^t.^{54, 1975,}^p.^1309.

[33] ^T.ELSTER, Gen. Rel. Grav. (in press).

[34] ^{P. V.}BLIOKH, ^{A. A.}MINAKOV, Astrophys. Sp. ^Sc.,^t.^{34, 1975,}^{p. L7.}

[35] ^H.^C.^OHANIAN,^{Int. J.}^Theor.Phys., ^t.9, 1974, p. 425.

[36] ^E.HERLT, ^H.STEPHANI, ^{Int. J.}^Theor.Phys., t. 15, 1976, p. 45; t. 17, 1978, p. 189.

[37] ^T.ELSTER, ^H.STEPHANI, Acta Phys. Pol., t. B 10, 1979, p. 1031.

[38] ^T.ELSTER, Astrophys. Sp. ^Sc.,^t.^71,1980, p. 171.

[39] ^R.^A.ISAACSON, Phys. Rev., ^t.166, 1968, p. 1272.

[40] ^{A. A.}MINAKOV, Astron. Zh., ^t.55, 1978, p. 966.

[41] P. ^V.BLIOKH, Â.Â.MINAKOV, Îzv.Vyssh. Ûcheb.^Zav.Radiofiz., ^t.21, 1978, p. 802.

[42] ^G.DAUTCOURT, ^K.FRITZE, ^Astron.Nachr., ^t.292, 1971, p. 211.

[43] ^J.PFARR, Gen. Rel. Grav., ^t.7, 1976, p. 459.

(Manuscrit reçu le 26 septembre ¹⁹⁸⁰⁾

Annales de l’Institut Henri Poincaré - Section A